# Intersection of two functions

Could you please try to find the intersections points of this two functions: $$y = x$$ and $$y=(a+x) \ {\rm e}^ {-2(a+x)}$$ with $$x\geq0$$

• Hi Ivan, Welcome to MSE! Can you please post the code you have tried. Sep 26, 2019 at 14:50
• Try the Lambert function en.wikipedia.org/wiki/Lambert_W_function Sep 26, 2019 at 14:55
– kglr
Sep 26, 2019 at 15:04
• The generalization of the Lambert function by Mező real.mtak.hu/39795/1/… is relevant Sep 26, 2019 at 15:13
• @yarchik Looks amazing. Sep 27, 2019 at 11:05

To visualize the pairs of real numbers {x, a} that satisfy the equation,

you can use ContourPlot:

ContourPlot[(a + x) E^(-2 (a + x)) == x, {x, -2, 1}, {a, 0, 5}]


or RegionPlot with options MeshFunctions + Mesh:

RegionPlot[True, {x, -2, 1}, {a, 0, 5},
BoundaryStyle -> None,
MeshFunctions -> {(#2 + #) E^(-2 (#2 + #)) - # &},
Mesh -> {{0}},
MeshStyle -> Directive[Thick, Red],
PlotStyle -> None,
PlotPoints -> 100]


Looks like you'll need some constraints on a.

a = -1;
f[x_] := (a + x) E^(-2 (a + x))
Plot[{x, f[x]}, {x, 0.001, 4}, PlotRange -> {Automatic, {-2, 1}}]